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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdet | Structured version Visualization version GIF version |
Description: Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
madjusmdet.b | ⊢ 𝐵 = (Base‘𝐴) |
madjusmdet.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) |
madjusmdet.d | ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) |
madjusmdet.k | ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) |
madjusmdet.t | ⊢ · = (.r‘𝑅) |
madjusmdet.z | ⊢ 𝑍 = (ℤRHom‘𝑅) |
madjusmdet.e | ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) |
madjusmdet.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
madjusmdet.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
madjusmdet.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
madjusmdet.j | ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
madjusmdet.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
madjusmdet | ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | madjusmdet.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
2 | madjusmdet.a | . 2 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
3 | madjusmdet.d | . 2 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | |
4 | madjusmdet.k | . 2 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | |
5 | madjusmdet.t | . 2 ⊢ · = (.r‘𝑅) | |
6 | madjusmdet.z | . 2 ⊢ 𝑍 = (ℤRHom‘𝑅) | |
7 | madjusmdet.e | . 2 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | |
8 | madjusmdet.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | madjusmdet.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | madjusmdet.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | |
11 | madjusmdet.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | |
12 | madjusmdet.m | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
13 | eqeq1 2739 | . . . 4 ⊢ (𝑘 = 𝑖 → (𝑘 = 1 ↔ 𝑖 = 1)) | |
14 | breq1 5151 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝐼 ↔ 𝑖 ≤ 𝐼)) | |
15 | oveq1 7438 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1)) | |
16 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑖 → 𝑘 = 𝑖) | |
17 | 14, 15, 16 | ifbieq12d 4559 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) |
18 | 13, 17 | ifbieq2d 4557 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
19 | 18 | cbvmptv 5261 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
20 | breq1 5151 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) | |
21 | 20, 15, 16 | ifbieq12d 4559 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) |
22 | 13, 21 | ifbieq2d 4557 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
23 | 22 | cbvmptv 5261 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
24 | eqeq1 2739 | . . . 4 ⊢ (𝑙 = 𝑗 → (𝑙 = 1 ↔ 𝑗 = 1)) | |
25 | breq1 5151 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝐽 ↔ 𝑗 ≤ 𝐽)) | |
26 | oveq1 7438 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 − 1) = (𝑗 − 1)) | |
27 | id 22 | . . . . 5 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗) | |
28 | 25, 26, 27 | ifbieq12d 4559 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)) |
29 | 24, 28 | ifbieq2d 4557 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
30 | 29 | cbvmptv 5261 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
31 | breq1 5151 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝑁 ↔ 𝑗 ≤ 𝑁)) | |
32 | 31, 26, 27 | ifbieq12d 4559 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)) |
33 | 24, 32 | ifbieq2d 4557 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
34 | 33 | cbvmptv 5261 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 23, 30, 34 | madjusmdetlem4 33791 | 1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 ≤ cle 11294 − cmin 11490 -cneg 11491 ℕcn 12264 ...cfz 13544 ↑cexp 14099 Basecbs 17245 .rcmulr 17299 CRingccrg 20252 ℤRHomczrh 21528 Mat cmat 22427 maDet cmdat 22606 maAdju cmadu 22654 subMat1csmat 33754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-dsmm 21770 df-frlm 21785 df-mat 22428 df-marrep 22580 df-subma 22599 df-mdet 22607 df-madu 22656 df-minmar1 22657 df-smat 33755 |
This theorem is referenced by: mdetlap 33793 |
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